RCOLGs

RCOLGs, short for recursive constrained ordered local graphs, is a theoretical class of graphs defined by a hierarchical, modular construction. In this framework, complex graphs are built from simpler building blocks called atoms, each with designated attachment points or ports. The construction proceeds in a recursive manner: at each step, an atom can be expanded by substituting its ports with subgraphs in a specified, finite order, connecting them according to a fixed attachment pattern. The resulting graph preserves a sense of locality, since all connections between components respect the prescribed ports and their ordering.

Construction and rules

An RCOLG begins with a set of base atoms, each a small graph with labeled ports. A rule specifies how to replace a port with another atom or subgraph and how to align and connect the ports of the attached subgraphs in a given sequence. This yields a tree-like decomposition of the overall graph, where the global structure emerges from the recursive composition of local pieces. The ordering constraints ensure that the sequence of connections is deterministic, which helps in defining canonical representations and comparing different RCOLGs.

Properties

RCOLGs emphasize modularity and locality, often enabling inductive reasoning about global properties from local components. Depending on the chosen atoms and attachment rules, RCOLGs can exhibit a range of structural features, such as bounded degree, planarity under compatible embeddings, or hierarchical depth measures. The class is flexible enough to model both simple hierarchical graphs and more intricate networks with repeated motifs.

Applications

Potential applications include modeling modular software architectures, circuit design, hierarchical network topologies, and biochemical or workflow pathways where repeated motifs and orderly assembly reflect real-world constraints.

See also

Modular graphs, hierarchical graphs, recursive graph classes, graph grammars.